Infinite Visits of a Rotating Line Through a Finite Point Set: A Combinatorial‑Geometric Proof — E8 Intelligence Research

FINDING: The 2011 IMO "windmill" problem demonstrates that a rotating line through a finite set of points in the plane will visit each point infinitely often, using a combinatorial‑geometric induction argument. | MATH: Let \(S=\{P_1,\dots,P_n\}\subset\mathbb{R}^2\) be in general position. For a pivot \(P_k\) and a line \(L\) through \(P_k\), define the rotation angle \(\theta(t)\) such that \(L(t)\) sweeps counter‑clockwise. The key relation is that the set of angles at which \(L(t)\) passes through any other point \(P_j\) is a finite, strictly increasing sequence modulo \(2\pi\). By induction on \(n\) and a "pivot‑switch" argument, one shows that for each \(P_j\) there exist infinitely many \(t\) with \(P_j\in L(t)\). | CONNECTION: No direct link to golden‑ratio or base‑60 ratios or crystallographic symmetry is evident in the problem or its solution. | DEPTH: 5 Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com